We prove that there is a randomized polynomial-time algorithm that given an edge-weighted graph $G$ excluding a fixed-minor $Q$ on $n$ vertices and an accuracy parameter $\varepsilon>0$, constructs an edge-weighted graph~$H$ and an embedding $\eta\colon V(G)\to V(H)$ with the following properties: * For any constant size $Q$, the treewidth of $H$ is polynomial in $\varepsilon^{-1}$, $\log n$, and the logarithm of the stretch of the distance metric in $G$. * The expected multiplicative distortion is $(1+\varepsilon)$: for every pair of vertices $u,v$ of $G$, we have $\mathrm{dist}_H(\eta(u),\eta(v))\geq \mathrm{dist}_G(u,v)$ always and $\mathrm{Exp}[\mathrm{dist}_H(\eta(u),\eta(v))]\leq (1+\varepsilon)\mathrm{dist}_G(u,v)$. Our embedding is the first to achieve polylogarithmic treewidth of the host graph and comes close to the lower bound by Carroll and Goel, who showed that any embedding of a planar graph with $\mathcal{O}(1)$ expected distortion requires the host graph to have treewidth $\Omega(\log n)$. It also provides a unified framework for obtaining randomized quasi-polynomial-time approximation schemes for a variety of problems including network design, clustering or routing problems, in minor-free metrics where the optimization goal is the sum of selected distances. Applications include the capacitated vehicle routing problem, and capacitated clustering problems.

翻译：我们证明存在一个随机多项式时间算法，给定边加权图$G$（排除固定极小元$Q$，顶点数为$n$）和精度参数$\varepsilon>0$，构造边加权图$H$及嵌入$\eta\colon V(G)\to V(H)$，满足以下性质： * 对于任意常数大小的$Q$，$H$的树宽是$\varepsilon^{-1}$、$\log n$及$G$中距离度规拉伸对数的多项式。 * 预期乘法失真为$(1+\varepsilon)$：对$G$中任意顶点对$u,v$，始终有$\mathrm{dist}_H(\eta(u),\eta(v))\geq \mathrm{dist}_G(u,v)$，且$\mathrm{Exp}[\mathrm{dist}_H(\eta(u),\eta(v))]\leq (1+\varepsilon)\mathrm{dist}_G(u,v)$。 我们的嵌入首次实现了宿主图树宽的多对数级别，并接近Carroll与Goel的下界：他们指出，对于平面图，任何预期失真为$\mathcal{O}(1)$的嵌入都要求宿主图的树宽为$\Omega(\log n)$。该嵌入还为无小类度规中优化目标为选定距离之和的多种问题（如网络设计、聚类或路由问题）提供了统一的随机拟多项式时间近似方案框架。应用场景包括带容量约束的车辆路径问题及带容量约束的聚类问题。